under the assumption that the BEC density \(\rho ^{(B)}\)  accounts for the DM density
in entirety. Consequently, for \(m \lesssim 6\,\) eV, one has \(\, T(a)< T_c(a) \,\forall \,a \,\) , where \(\, T(a) = 2.7 \, a^{-1} \,\)  is the ambient temperature
of the universe. This therefore points to the formation of a BEC of the bosons
in the early phase of evolution of the universe, once the BEC density is taken
to be of the order of the critical density of the universe at the present
epoch
Note however the difference between the expression (REF )
and that derived in [1]}. This is because the former is obtained upon
considering the BEC of neutral bosons, which are their own anti-particles, with
no charge or other hidden quantum numbers and a vanishing chemical potential
(e.g. gravitons or axions). In contrast, the authors in [1]} have
considered charged bosons with a non-vanishing chemical potential, distinct
particles and anti-particles and a net BEC charge..
